Hyperk\"ahler Isometries Of K3 Surfaces

We consider symmetries of K3 manifolds. Holomorphic symplectic automorphisms of K3 surfaces have been classified, and observed to be subgroups of the Mathieu group $M_{23}$. More recently, automorphisms of K3 sigma models commuting with $SU(2)\times SU(2)$ $R$-symmetry have been classified by Gaberdiel, Hohenegger, and Volpato. These groups are all subgroups of the Conway group. We fill in a small gap in the literature and classify the possible hyperk\"ahler isometry groups of K3 manifolds. There is an explicit list of $40$ possible groups, all of which are realized in the moduli space. The groups are all subgroups of $M_{23}$.